Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dva1dim | Structured version Visualization version GIF version |
Description: Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) 𝐹 whose trace is 𝑃 rather than 𝑃 itself; 𝐹 exists by cdlemf 34869. 𝐸 is the division ring base by erngdv 35299, and 𝑠‘𝐹 is the scalar product by dvavsca 35323. 𝐹 must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.) |
Ref | Expression |
---|---|
dva1dim.l | ⊢ ≤ = (le‘𝐾) |
dva1dim.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dva1dim.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dva1dim.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dva1dim.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dva1dim | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dva1dim.h | . . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dva1dim.t | . . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | dva1dim.e | . . . . . . . . . 10 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | tendocl 35073 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇) |
5 | dva1dim.l | . . . . . . . . . 10 ⊢ ≤ = (le‘𝐾) | |
6 | dva1dim.r | . . . . . . . . . 10 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
7 | 5, 1, 2, 6, 3 | tendotp 35067 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)) |
8 | 4, 7 | jca 553 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))) |
9 | 8 | 3expb 1258 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))) |
10 | 9 | anass1rs 845 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))) |
11 | eleq1 2676 | . . . . . . 7 ⊢ (𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ↔ (𝑠‘𝐹) ∈ 𝑇)) | |
12 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑔 = (𝑠‘𝐹) → (𝑅‘𝑔) = (𝑅‘(𝑠‘𝐹))) | |
13 | 12 | breq1d 4593 | . . . . . . 7 ⊢ (𝑔 = (𝑠‘𝐹) → ((𝑅‘𝑔) ≤ (𝑅‘𝐹) ↔ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))) |
14 | 11, 13 | anbi12d 743 | . . . . . 6 ⊢ (𝑔 = (𝑠‘𝐹) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) ↔ ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))) |
15 | 10, 14 | syl5ibrcom 236 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → (𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)))) |
16 | 15 | rexlimdva 3013 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)))) |
17 | simpll 786 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
18 | simplr 788 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → 𝐹 ∈ 𝑇) | |
19 | simprl 790 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → 𝑔 ∈ 𝑇) | |
20 | simprr 792 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → (𝑅‘𝑔) ≤ (𝑅‘𝐹)) | |
21 | 5, 1, 2, 6, 3 | tendoex 35281 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) → ∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔) |
22 | 17, 18, 19, 20, 21 | syl121anc 1323 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → ∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔) |
23 | eqcom 2617 | . . . . . . 7 ⊢ ((𝑠‘𝐹) = 𝑔 ↔ 𝑔 = (𝑠‘𝐹)) | |
24 | 23 | rexbii 3023 | . . . . . 6 ⊢ (∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔 ↔ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)) |
25 | 22, 24 | sylib 207 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)) |
26 | 25 | ex 449 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) → ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹))) |
27 | 16, 26 | impbid 201 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹) ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)))) |
28 | 27 | abbidv 2728 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))}) |
29 | df-rab 2905 | . 2 ⊢ {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))} | |
30 | 28, 29 | syl6eqr 2662 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 {crab 2900 class class class wbr 4583 ‘cfv 5804 lecple 15775 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 TEndoctendo 35058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-undef 7286 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 |
This theorem is referenced by: dvhb1dimN 35292 dia1dim 35368 |
Copyright terms: Public domain | W3C validator |